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Question

Noah noticed that an air conditioner from Company A can cool a hall in 10 minutes, while an air conditioner from Company B can do the same in 15 miutes.
It is a very hot day, so Noah has turned on both the air conditioners.
Find how long both the air conditioners will take to cool the hall.

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Solution

\(Company~ A ~air~conditioner:\)
Time taken to cool the hall = 10 minutes
Work done in 1 minute = \(\dfrac{1}{10}\)

\(Company~B~air~conditioner:\)
Time taken to cool the hall = 15 minutes
Work done in 1 minute = \(\dfrac{1}{15}\)

\(Both~air~ conditioners:\)
Total work done in 1 minute \(= \dfrac{1}{10} +\dfrac{1}{15}\)

= \(\dfrac{15~+~10}{15~\times~10}\)

\(= \dfrac{25}{150}\)

\(= \dfrac{1}{6}\)

The time taken to cool the hall will be the inverse of the work done in 1 minute.
Thus,
Time taken to cool the hall = 6 minutes

Both the air conditioners together will take 6 minutes to cool the hall.

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